Bulk Fermi surface of the Weyl type-II semimetallic candidate $\gamma -{\mathrm{MoTe}}_2$

The electronic structure of semimetallic transition-metal dichalcogenides, such as ${\mathrm{WTe}}_2$ and orthorhombic $\gamma -{\mathrm{MoTe}}_2$, are claimed to contain pairs of Weyl points or linearly touching electron and hole pockets associated with a nontrivial Chern number. For this reason, these compounds were recently claimed to conform to a new class, deemed type-II, of Weyl semimetallic systems. A series of angle-resolved photoemission experiments (ARPES) claim a broad agreement with these predictions detecting, for example, Fermi arcs at the surface of these crystals. We synthesized single crystals of semimetallic ${\mathrm{MoTe}}_2$ through a Te flux method to validate these predictions through measurements of its bulk Fermi surface (FS) via quantum oscillatory phenomena. We find that the superconducting transition temperature of $\gamma -{\mathrm{MoTe}}_2$ depends on disorder as quantified by the ratio between the room- and low-temperature resistivities, suggesting the possibility of an unconventional superconducting pairing symmetry. Similarly to ${\mathrm{WTe}}_2$, the magnetoresistivity of $\gamma -{\mathrm{MoTe}}_2$ does not saturate at high magnetic fields and can easily surpass ${10}^6$%. Remarkably, the analysis of the de Haas–van Alphen (dHvA) signal superimposed onto the magnetic torque indicates that the geometry of its FS is markedly distinct from the calculated one. The dHvA signal also reveals that the FS is affected by the Zeeman effect precluding the extraction of the Berry phase. A direct comparison between the previous ARPES studies and density-functional-theory (DFT) calculations reveals a disagreement in the position of the valence bands relative to the Fermi level ${\varepsilon }_F$. Here, we show that a shift of the DFT valence bands relative to ${\varepsilon }_F$, in order to match the ARPES observations, and of the DFT electron bands to explain some of the observed dHvA frequencies, leads to a good agreement between the calculations and the angular dependence of the FS cross-sectional areas observed experimentally. However, this relative displacement between electron and hole bands eliminates their crossings and, therefore, the Weyl type-II points predicted for $\gamma -{\mathrm{MoTe}}_2$.

Figure 1

(a) Resistivity $\rho$, for currents flowing along the $a\mathrm{axis}$, as a function of the temperature $T$ for three representative single crystals displaying resistivity ratios $\rho \left(300\text{K}\right)/\rho \left(2\right)\text{K}$ between 380 and $\gtrsim 1000$. (b) $\rho$ as a function of $T$ for each single crystal indicating that $T_c$ depends on sample quality: it increases from an average transition middle point value of $\sim 130$ mK for the sample displaying the lowest ratio to $\sim 435$ mK for the sample displaying the highest one. The apparent hysteresis is due to a nonideal thermal coupling between the single crystals, the heater, and the thermometer. (c) $\rho$ as a function of the field ${\mu }_{0}H$ applied along the $c$ axis at a temperature $T=1.7$ K for a fourth crystal characterized by a resistivity ratio of $\sim 207$. Notice (i) the nonsaturation of $\rho \left({\mu }_{0}H\right)$ and (ii) that $\mathrm{\Delta }\rho \left({\mu }_{0}H\right)/{\rho }_0=[\rho \left({\mu }_{0}H\right)-{\rho }_0]/{\rho }_0$, where ${\rho }_0=\rho ({\mu }_{0}H=0\text{T},T=2\text{K})$ surpasses $1.4\times {10}^6$% at ${\mu }_{0}H=60$ T. (d) $\rho$ as a function of ${\mu }_{0}H$ applied along the $b\mathrm{axis}$ also at $T=1.7$ K and for the same single crystal. (e) Shubnikov–de Haas signal superimposed onto the magnetoresistivity for ${\mu }_{0}H\parallel c\mathrm{axis}$ and for three temperatures $T=8$ K (blue line), 4.2 K (red line), and 1.7 K (black line), respectively. (f) Oscillatory signal (black line) superimposed onto the magnetic susceptibility $\mathrm{\Delta }\chi =\partial (\tau /{\mu }_{0}H)/\partial {\mu }_{0}H$, where $\tau$ is the magnetic torque.

Figure 2

(a) Typical de Haas–van Alphen (red trace) and Shubnikov–de Haas (black trace) signals superimposed onto the magnetic torque and the magnetoresistivity, respectively, for fields aligned along the $c$ axis of $\gamma -{\mathrm{MoTe}}_2$ single crystals at $T\simeq 30$ mK. The same panel shows the FFT spectra for each signal revealing just two main frequencies or FS cross-sectional areas. (b) dHvA and SdH signals and corresponding FFT spectra obtained from the same single crystals for $H$ aligned nearly along the $a$ axis of each single crystal. (c), (d) Amplitude of the peaks observed in the FFT spectra for $H\parallel c$ axis and as a function of $T$ including the corresponding fits to the LK formula from which the effective masses are extracted. (e), (f) Amplitude of two representative peaks observed in the FFT spectra for $H\parallel a$ axis and as a function of $T$ with the corresponding fits to the LK formula to extract their effective masses.

Figure 3

(a) Electronic band structures of $\gamma -{\mathrm{MoTe}}_2$ calculated with and without the inclusion of spin-orbit coupling (SOC). These calculations are based on the crystallographic structure measured at $T=100$ K. (b) Comparison between the calculated electronic bands based upon the crystallographic structures measured at $T=100$ and 12 K, respectively. (c), (d) Respectively, side and top views of the calculated FS. (e)–(h) Fermi surface sheets resulting from electron bands. Notice the marked two-dimensional character of several of the electronlike FS sheets. (i), (k), (l) Holelike sheets around the $\mathrm{\Gamma }\mathrm{point}$.

Figure 4

(a) Angular dependence of the FS cross-sectional areas or, through the Onsager relation, de Haas–van Alphen frequencies with respect to the main crystallographic axes. (b) Experimentally observed dHvA spectra as a function of the frequency $F$ for several angles $\theta$ between the $c$ and and the $b$ axes and for several values of the angle $-\varphi$ between the $c$ and the $a\mathrm{axes}$.

Figure 5

(a) ARPES energy distribution map along $k_y$ for $k_x=0$ from Ref. [23]. (b) Derivative of the energy distribution map. Vertical blue lines indicate the diameter of the observed hole pocket, with its area corresponding to a frequency $F\sim 0.33$ kT. (c) Ribbons obtained from our DFT calculations showing the $k_z$ projection for all bulk bands which are plotted along the same direction as the ARPES energy distribution map in (a). Different colors are chosen to indicate distinct bands. Purple dotted line corresponds to the original position of the Fermi level ${\epsilon }_F$ according to DFT, while the white one corresponds to the position of ${\epsilon }_F$ according to ARPES. Notice that the ARPES bands are shifted by $\sim -50$ meV with respect to the DFT ones. (d) ARPES energy distribution map corresponding to a cut along the $k_x\mathrm{direction}$ with $k_y=0$, from Ref. [21]. Vertical red dotted lines indicate the diameter of the observed electron pockets or $\sim 0.2{\AA }^{-1}$. SS stands for “surface state.” (e) DFT ribbons showing the $k_z$ projection for all bulk bands plotted along the same direction as the ARPES EDM in (d). In both panels, black lines depict the original position of ${\epsilon }_F$ while the yellow lines are guides to the eyes illustrating the difference in energy between the top of the deepest DFT calculated hole band and its equivalent according to ARPES.

Figure 6

(a) Experimentally observed dHvA spectra as a function of $F$ for several angles $\theta$ and $-\varphi$, where the magenta and the blue lines act as guides to the eyes and as identifiers of, respectively, electronlike and holelike orbits according to the shifted band structure. Clear blue line depicts a possible hole orbit associated with very small peaks in the FFT spectra. (b) Angular dependence of the dHvA orbits, or frequencies on the FS resulting from the shifted bands in absence of spin-orbit coupling, where magenta and blue markers depict electronlike and holelike orbits on the FS, respectively. Notice the qualitative and near quantitative agreement between the calculations and the experimental observations. (c) Angular dependence of the dHvA frequencies for the shifted electron and hole bands in the presence of spin-orbit coupling. Here, electron orbits are depicted by magenta and orange markers while the hole ones are indicated by blue and clear blue markers. The experimental data are better described by the non-spin-orbit split bands.

Figure 7

(a) Electronic band structure calculated with the inclusion of SOC after the holelike bands have been shifted by $-50$ meV and the electron ones by $+35$ meV with the goal of reproducing the observed dHvA frequencies and their angular dependence. Notice that these shifts suppress the crossings between the electron and hole bands and therefore the Weyl type-II points of the original band structure. (b) Fermi surface resulting from these band shifts. (c) FS top view. (d), (e) Electronlike FS sheets. (f), (g) Holelike sheets.